| L(s) = 1 | + 3·2-s − 3-s + 6·4-s − 5-s − 3·6-s + 10·8-s − 3·10-s + 6·11-s − 6·12-s + 15-s + 15·16-s + 7·17-s − 4·19-s − 6·20-s + 18·22-s + 4·23-s − 10·24-s − 8·25-s − 27-s + 5·29-s + 3·30-s − 7·31-s + 21·32-s − 6·33-s + 21·34-s + 6·37-s − 12·38-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 0.577·3-s + 3·4-s − 0.447·5-s − 1.22·6-s + 3.53·8-s − 0.948·10-s + 1.80·11-s − 1.73·12-s + 0.258·15-s + 15/4·16-s + 1.69·17-s − 0.917·19-s − 1.34·20-s + 3.83·22-s + 0.834·23-s − 2.04·24-s − 8/5·25-s − 0.192·27-s + 0.928·29-s + 0.547·30-s − 1.25·31-s + 3.71·32-s − 1.04·33-s + 3.60·34-s + 0.986·37-s − 1.94·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(19.22705458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.22705458\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + T^{2} + 2 T^{3} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + T + 9 T^{2} + 12 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 35 T^{2} - 112 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 - 7 T + 59 T^{2} - 230 T^{3} + 59 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 29 T^{2} + 120 T^{3} + 29 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 49 T^{2} - 120 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 5 T + 89 T^{2} - 280 T^{3} + 89 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 7 T + 101 T^{2} + 426 T^{3} + 101 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 43 | $S_4\times C_2$ | \( 1 + 9 T + 81 T^{2} + 542 T^{3} + 81 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 53 | $S_4\times C_2$ | \( 1 + 7 T + 113 T^{2} + 632 T^{3} + 113 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 2 T + 127 T^{2} - 336 T^{3} + 127 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 13 T + 209 T^{2} - 1596 T^{3} + 209 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 22 T + 267 T^{2} - 2368 T^{3} + 267 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 7 T + 109 T^{2} - 666 T^{3} + 109 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 28 T + 447 T^{2} - 4600 T^{3} + 447 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 19 T + 333 T^{2} - 3130 T^{3} + 333 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 14 T + 295 T^{2} - 2304 T^{3} + 295 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 23 T + 419 T^{2} + 4330 T^{3} + 419 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 25 T + 475 T^{2} - 5254 T^{3} + 475 p T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.37752997701067641948445898943, −6.78878321813306502837326791991, −6.78743860200283861596394540524, −6.77799594177129029489483589717, −6.41490450373664739078251397637, −6.29246873445307007671852935001, −5.98890164507959418644104762611, −5.41722069970985333566563500053, −5.39678971557781539100590285406, −5.34231240505366626727734684273, −5.10553567378500718827325523978, −4.59134915139548504306165877605, −4.38294996475111405008141994021, −4.05629103304253146510315439106, −3.85974523882676149186579881171, −3.69459745816230411757482905338, −3.44916452347004350568867690768, −3.18401666713030731483646301308, −2.85941479972507689839230260348, −2.27593158049197524015797395717, −2.10526180316599979259791674967, −1.78198599507577835701458639944, −1.41086861196777148851246708092, −0.74804007563192242623043442198, −0.69616036685243516737383583317,
0.69616036685243516737383583317, 0.74804007563192242623043442198, 1.41086861196777148851246708092, 1.78198599507577835701458639944, 2.10526180316599979259791674967, 2.27593158049197524015797395717, 2.85941479972507689839230260348, 3.18401666713030731483646301308, 3.44916452347004350568867690768, 3.69459745816230411757482905338, 3.85974523882676149186579881171, 4.05629103304253146510315439106, 4.38294996475111405008141994021, 4.59134915139548504306165877605, 5.10553567378500718827325523978, 5.34231240505366626727734684273, 5.39678971557781539100590285406, 5.41722069970985333566563500053, 5.98890164507959418644104762611, 6.29246873445307007671852935001, 6.41490450373664739078251397637, 6.77799594177129029489483589717, 6.78743860200283861596394540524, 6.78878321813306502837326791991, 7.37752997701067641948445898943
